Simplify the following expression: $y = \dfrac{-7x^2+22x+24}{-7x - 6}$
Explanation: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-7)}{(24)} &=& -168 \\ {a} + {b} &=& &=& {22} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-168$ and add them together. Remember, since $-168$ is negative, one of the factors must be negative. The factors that add up to ${22}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-6}$ and ${b}$ is ${28}$ $ \begin{eqnarray} {ab} &=& ({-6})({28}) &=& -168 \\ {a} + {b} &=& {-6} + {28} &=& 22 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({-7}x^2 {-6}x) + ({28}x +{24}) $ Factor out the common factors: $ x(-7x - 6) - 4(-7x - 6)$ Now factor out $(-7x - 6)$ $ (-7x - 6)(x - 4)$ The original expression can therefore be written: $ \dfrac{(-7x - 6)(x - 4)}{-7x - 6}$ We are dividing by $-7x - 6$ , so $-7x - 6 \neq 0$ Therefore, $x \neq -\frac{6}{7}$ This leaves us with $x - 4; x \neq -\frac{6}{7}$.